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Physics Letters B 783 (2018) 253–262

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Conformal symmetry: Towards the link between the Fermi and the

Planck scales

Mikhail Shaposhnikov

, Andrey Shkerin

a,b,∗

Institute of Physics, Laboratory for Particle Physics and Cosmology, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland

Institute for Nuclear Research of the Russian Academy of Sciences, 60th October Anniversary prospect 7a, 117312, Moscow, Russia

a r t i c l e i n f o a b s t r a c t

Article history:

Received

28 March 2018

Received

in revised form 25 June 2018

Accepted

29 June 2018

Available

online 3 July 2018

Editor: A.

Ringwald

If the mass of the Higgs boson is put to zero, the classical Lagrangian of the Standard Model (SM)

becomes conformally invariant (CI). Taking into account quantum non-perturbative QCD effects violating

CI leads to electroweak symmetry breaking with the scale v ∼ 

QCD

∼ 100 MeV which is three orders of

magnitude less than it is observed experimentally. Depending on the mass of the top quark, the radiative

corrections may lead to another minimum of the effective potential for the Higgs ﬁeld with v  M

where M

is the Planck mass, at least 16 orders of magnitude more than it is observed. We explore

yet another source of CI breaking associated with gravity. We suggest a non-perturbative mechanism

that can reproduce the observed hierarchy between the Fermi and the Planck scales, by constructing an

instanton conﬁguration contributing to the vacuum expectation value of the Higgs ﬁeld. The crucial role

in this effect is played by the non-minimal coupling of the Higgs ﬁeld to the Ricci scalar and by the

approximate Weyl invariance of the theory for large values of the Higgs ﬁeld.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction and setup

The observed value of the Electroweak (EW) scale, represented

by the Standard Model (SM) Higgs boson mass m

, poses a chal-

lenge

the physics beyond the SM must deal with. One aspect of

this challenge comes from the fact that radiative corrections to m

engendered by the presence of degrees of freedom with some

heavy mass scale (existing, for instance, in the standard GUTs –

Grand Uniﬁed theories), shift it towards that scale. The apparent

stability of the Higgs mass against such corrections requires ei-

ther

ﬁne-tuning among the parameters of the theory or a special

mechanism of their systematic suppression. The second aspect of

the problem concerns with the smallness of the ratio of the EW

scale to the GUT scale or to the Planck scale at which quantum

gravity effects are expected to come into play. Combined together,

the two issues are known as the hierarchy problem (for reviews

see, e.g. [1,2]).

The

ﬁrst part of the hierarchy problem, related to the stability

of the EW scale against perturbative quantum corrections, does not

Corresponding author.

E-mail

addresses: Mikhail.Shaposhnikov@epﬂ.ch (M. Shaposhnikov),

Andrey.Shkerin@epﬂ.ch (A. Shkerin).

appear in theories containing light particles only,

for detailed ar-

guments

and previous references see [4–6]. Of course, even if one

evades any heavy mass thresholds coming with the physics beyond

the SM, the Planck scale enters unavoidably any theory comprising

the SM and General Relativity (GR). As the Planck mass M

does

not represent a mass of any particle but rather serves as a pa-

rameter

measuring the strength of gravitational interaction (for an

overview see, e.g. a relevant chapter in [7]), the argument may re-

main

in force in the presence of gravity as well [5,6,8].

is tempting to use the conformal symmetry for solution of

the second part of the hierarchy problem [9]. Indeed, since at

the classical level the SM Lagrangian acquires a conformal invari-

ance

(CI) once m

is put to zero, one can imagine to start from

the conformally-invariant classical SM which has no EW symmetry

breaking and generate the Higgs mass due to the CI violation.

One

of the possible ways to generate the Higgs mass within the

CI setting is associated with quantum conformal anomaly (see, e.g.,

[10]). Indeed, the UV regularisation of renormalizable ﬁeld theories

necessarily introduces a parameter with the dimension of mass,

which violates CI at the quantum level and thus makes it to be

This requirement does not exclude Grand Uniﬁed Theories, which can be con-

structed

without leptoquarks [3].

https://doi.org/10.1016/j.physletb.2018.06.068

0370-2693/

SCOAP

254 M. Shaposhnikov, A. Shkerin / Physics Letters B 783 (2018) 253–262

anomalous.

As a result, the effective potential for the Higgs ﬁeld,

accounting for higher-order radiative corrections, may develop a

minimum displaced from the origin, potentially leading to the vac-

uum

expectation value (vev) v of the Higgs ﬁeld small compared

with M

or the GUT scale [13,14].

The

Coleman–Weinberg (CW) scenario [13]in the SM can in-

deed

be realised [15–17], but it leads to the Higgs and the top

quark masses m

 7GeV and m

 80 GeV being far from those

observed experimentally. If we take the physical values of dimen-

sionless

Higgs self-coupling λ and top quark Yukawa coupling y

the effective potential of the CI SM has a minimum around the

QCD scale 

QCD

∼ 100 MeV, associated with conﬁnement and

quark condensates [18], which is too far from the one realised in

Nature. If the value of the top quark Yukawa coupling is smaller

than some critical value, y

< y

crit

, this minimum is unique. For

> y

crit

yet another minimum is generated by the CW mecha-

nism

[19], with the very large vev v  M

, now many orders of

magnitude larger that the EW scale. Due to experimental uncer-

tainties

it is not known yet whether y

is larger or smaller than

crit

(for an overview see, e.g., [20]), but in any event the predic-

tions

of the CI SM are in sharp contrast with experiment.

spite of this failure, the no-scale CI theories look very at-

tractive

and motivated many authors to search for different ex-

tensions

of the SM, in which the mechanism may work and be

phenomenologically acceptable. We mention just a few. The ex-

tended

scalar sector was discussed on general grounds in [21],

more recent works deal with the SM extended by right-handed

neutrinos and a real scalar ﬁeld [22], by an Abelian B − L gauge

ﬁeld [23–25], or by non-abelian gauge groups [26,27].

All

the considerations of the CI theories up to date were carried

out without gravity. There is a clear rationale for this, based on

(nearly scale-invariant) perturbation theory [28]: any perturbative

corrections

to the effective potential of the Higgs ﬁeld coming from

gravity are suppressed by the Planck mass [29], and in the absence

of heavy particles they are numerically small. In the SM, the largest

contribution is of the order y

, which is negligible at the

weak scale.

Clearly,

the general diﬃculty of working with quantum gravity

is that we are not aware of an explicit UV completed theory reduc-

ing

to (CI) SM and GR at low energy scales. What one can do in

this situation is to attribute to the unknown UV physics the prop-

erties

which are found to be useful in resolving apparently low

energy physic’s questions. In particular, one can imagine that the

Higgs mass arises purely due to some quantum gravity effects. Hav-

ing

accepted the CI setting, this could only imply the existence of

a non-perturbative mechanism driving the Higgs ﬁeld vev towards

its observed value, some 17 orders of magnitude below M

The

aim of this paper is to argue that gravity can indeed gen-

erate

in a non-perturbative way a new mass scale, many orders of

magnitudes smaller than the Planck scale. Non-perturbative phe-

nomena

can manifest themselves in various ways. As one exam-

ple,

they can be associated with the strong coupling scale around

which the content of the theory is reorganized, and, in particular,

the physical degrees of freedom are rearranged.

Another possi-

bility

is the existence of euclidean classical conﬁgurations — the

instantons — that contribute to correlation functions of the theory

and may eventually result in drastic changes of the low energy ob-

If the requirement of renormalisability is removed, the theory can be made con-

formal

at the quantum level [8,11,12]. We will consider such theories in a separate

publication.

For discussion of this possibility in context with the hierarchy problem see, e.g.,

[30].

servables.

In this paper our main concern will be with the second

effect. In fact, it is not diﬃcult to come to an idea that instantons

or, more precisely, their large actions may somehow be involved in

generating the hierarchy of scales. The smallness of the ratio v/M

can be viewed as a result of an exponentially strong suppression

of the Planck scale that generally appears in physical quantities as

the only classical scale of the theory. Our intention is to use some

simple yet explicit models comprising the Higgs and the gravita-

tional

ﬁelds as a playground in which the existence conditions for

such instantons can be studied.

course, lacking a UV complete theory encompassing the SM

and GR brings irremovable ambiguities in our analysis. Neverthe-

less,

we argue that this ambiguity does not make quantitative

investigation meaningless once it is clearly stated under what as-

sumptions

about the high energy behavior of the theory we work.

We ﬁnd this kind of approach quite appealing, as the line of rea-

soning

can be reverted easily by claiming the observed value of v

be an argument in favour of those properties of the unknown

theory, for which the working mechanism of generation of the

Higgs mass is found.

Let

us specify the framework in which we will work. Following

the discussion above, we would like to exclude from considera-

tion

possible quantum (perturbative) corrections to the Higgs ﬁeld

vev, coming with the heavy mass thresholds associated with new

physics. To this end, we require no degrees of freedom with the

mass scales above the EW scale appear in the theory. That is, we

demand the only classical dimensional parameter in the theory be

the Planck mass. In this case, the Higgs-gravity sector of a theory

under investigation, which we will be mainly interested in, is gov-

erned

by the Horndeski construction or its extensions [32,33]. The

vastness of possible models is further restricted by the require-

ment

to reproduce the SM Higgs sector and GR at low energies

and by the assumption that among higher-dimensional operators

activating at high energies those are present that we ﬁnd useful

for the purpose of generating the hierarchy of scales.

It should be

stressed that our goal here is to ﬁnd an example of a model in

which a mechanism of an exponential suppression of the Planck

mass due to instantons exists. Hence, we do not have an intention

to perform a survey of all possible models by studying effects from

all possible higher-dimensional operators containing the Higgs and

gravity ﬁelds. Nor do we intend to argue that a toy theory chosen

to illustrate the mechanism can indeed be consistently embedded

into the UV complete theory of gravity.

will ﬁnd that the crucial ingredient of the theory admitting

the instantons with the desired properties is the non-minimal cou-

pling

of the Higgs ﬁeld to the Ricci scalar. We will also ﬁnd that

the instantons generating the large hierarchy of scales favour the

(approximate) Weyl invariance of the theory for large values of the

scalar ﬁeld.

The

paper is organized as follows. In sec. 2 we introduce a sim-

ple

model describing the dynamics of the gravitational and the

classically massless scalar ﬁelds. We analyze euclidean classical

conﬁgurations arising in this model, and discuss their possible in-

ﬂuence

on the vev of the scalar ﬁeld. The results of the analysis

motivate us to introduce certain modiﬁcations into the model. In

sec. 3 we incorporate these modiﬁcations step by step. We ﬁnd

that the contribution of a certain classical conﬁguration (the sin-

gular

instanton) to the vev of the scalar ﬁeld can actually make

the latter non-zero and, at the same time, many orders of magni-

tude

smaller than the Planck scale. In sec. 4 we apply our ﬁndings

Perhaps, the most instructive example of this phenomenon is a discrete sym-

metry

restoration in quantum mechanics of one dimension [31].

The structure of theories at high energies can also be subject to constraints, e.g.,

by the requirement of asymptotic safety [34].

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